where \(Q^{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  is a cofibrant replacement functor in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  with natural weak equivalence \(q \colon Q^{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\rightarrow 1\) , and where the second morphism \(e_{|F}\)  is the component at \(F\)  of the evaluation of the adjunction \(\tilde{\mathsf {c}} \dashv \operatorname{ev}_*\) .
Since every object in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  is cofibrant and since \(\tilde{\mathsf {c}}\)  is left Quillen, we have that \(\tilde{\mathsf {c}}(q_{|\operatorname{ev}_* F})\)  is a weak equivalence.
Further, the morphism \(e_{|F}\)  is an equivalence by Lemma REF .
The claim now follows by [1]}.
